Donaldson theory is the study of smooth 4-manifold

4-manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...

s using gauge theory

Gauge theory

In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

. It was started by Simon Donaldson

Simon Donaldson

Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...

 (1983) who proved Donaldson's theorem

Donaldson's theorem

In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...

 restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold.

Most of the results of Donaldson theory depend on the manifold having a differential structure, and are false for topological 4-manifolds.

Many of the theorems in Donaldson theory can now be proved more easily using Seiberg–Witten theory.